Multiple-beam interference
We first consider diffraction by a single slit and by a double slit. After passing a slit, the light beam widens. It widens the more, the narrower the slit is as compared to the wavelength.
 |
Diffraction on a single slit. Top: monochromatic light, bottom: white light. In the following we assume that the slits are very narrow and only look at what happens within the central bright region. The dark bands in the left picture would be far outside of the details shown in the following images.
|
 |
Diffraction by a double slit
A plane wave coming in from the left falls on an absorbing screen with two narrow slits. The outgoing waves on the right show alternating constructive and destructive interference. Double-click on the picture will switch on the animation, a single click switches it off again.
|
| Diffraction pattern of a double slit. Top: monochromatic light. This pattern corresponds to the animated picture above.
White light produces a coloured pattern (which in most experimental situations is rather dim and therefore the colours are not clearly seen).
|
 | A simple home experiment: Diffraction by a double hole. Two tiny holes were pierced with a sewing needle into an aluminium wrapping foil. Looking through these holes at a "point-like" light source, one can see a diffraction image similar to that shown here. For this image, the pierced foil has been fastened before the front lens of a digital camera, the light source was the reflection of a halogen lamp in a small silvery glass ball (as used for christmas-tree decoration). The diameter of the holes was 0.07 mm, the centre-to-centre distance 0.3 mm.
|
 |
Three slits. Adding a third slit at the same distance, the intensity maxima are preserved. Between them there are now two dark bands and a faint secondary maximum.
The pattern produced by white light shows more intense colours.
|
 |
Seven equidistant slits: the bright "lines" are now separated by six dark bands and five faint secondary maxima.
The colours of the "first order" diffraction maximum on both sides of the central white beam approach saturated spectral colours. The short-wave (blue-violet) end of the third-order diffraction already overlaps the long-wave (red) end of the second order.
|
 |
The more slits, the narrower are the main maxima and the weaker are the secondary maxima between them. (Here: 15 slits.)
|
Optical diffraction gratings as used for spectrometers and spectroscopes are something like arrays of hundreds of slits. To make use of the obtainable high resolution, additional optical elements are necessary: a collimating lens to convert the divergent rays of the point source to a beam of parallel rays (plane waves), and a telescope to view the spectrum enlarged.
An array of reflecting stripes separated by mat areas has the same effect in reflected light as a transparent grating in transmission.
Examples
 |
Everybody knows the vivid colours seen on compact discs. There is a mirror image showing no colour-splitting, but depending on the viewing angle, diffraction images of first or higher order can be seen. In case of small light sources, in first order almost pure spectral colours appear. |
 |
Under favourable conditions (dark background, but against the light) also spider webs may shine in vivid colours. For that, there are different possible reasons. In the picture shown, the main effect is due to the regular array of the glittering tiny sticky droplets on the capture threads. The web is shown approximately in natural size, intentionally slightly out of focus.
The pages "spider webs" present more images and give more detailed explanations. |
 |
If instead of simple stripes a more involved pattern is produced, colourful images may result, as in the case of holograms. Best known examples probably are those seen on bank-notes.
|
 |
Birds' feathers consist of a hollow shaft with branches fused to it on opposite sides. The branches (barbs) carry barbules which in the feathers' vanes are so small, dense, and regularly ordered that they form a diffraction grating. The picture on the left shows the feather of a pigeon, illuminated by a white LED in the background. Below at the left an enlarged detail where the focussing was on the feather, at the right the same with focus on the light source in the background. |
 |
|
Iridescent colours
Only in exceptional cases the colours of natural objects are due to regular arrays of shiny or transparent areas side by side, separated by mat or opaque areas. Quite frequent are interference colours due to more or less regularly stacked layers or other structures differing in their refractive index, so that they reflect light. Few transparent layers separated by small gaps produce vivid colours in reflected light.
| Colours of a thin dielectric layer in reflected light before a dark background. These are the same colours which we know from soap films. The scale gives the optical path difference of the two rays reflected on the upper and the lower surface.
|
 |
Two transparent layers, separated by an air gap of the same optical thickness. The scale gives the optical distance between the front surfaces of the two layers. (Optical path length means the path length multiplied by the refractive index of the medium.)
|
 |
Four transparent layers separated by air gaps. The more layers, the narrower the maxima of reflected light and, correspondingly, the more saturated the colours.
|
Examples
Mother-of-pearl consists of thin aragonite tiles held together by layers of organic matter (conchioline). The aragonite tiles, in the case of Haliotis, are about 500 nm thick (T.L. Tan, D. Wong, and P. Lee: Optics Express 12 (2004), No. 20 p. 4847). Thus the layers are relatively thick, resulting in less saturated colours. The high reflectivity is also due to the layered structure, like that of a stack of transparent foils.
The shell of an abalone (Haliotis), a marine gastropod. Left: Inner side of the shell of 15 cm diameter (6 in), right: enlarged detail.
Labradorite, a feldspar mineral of the plagioclase series (NaAlSi3O8 to CaAl2Si2O8), is grey, transparent, and exhibits iridescent colours. This is due to the laminar structure of the crystals (unmixing lamellae of albite NaAlSi3O8 and anorthite CaAl2Si2O8). The two components have slightly different refractive indices. This leads to very small reflection of light at the boundaries, but the large number compensates the smallness. As in the case of soap bubbles, the colour seen depends on the thickness of the laminae and the viewing angle.
A polished piece of labradorite from Madagascar, viewed under two different angles.
Meat: Striated muscles consist of fibers. The striae visible under the microscope reveal a regular layered structure, going along with periodic changes of the refractive index. Therefore, sometimes iridescent colours also can be seen in meat.
Pieces of ham. The left image is in natural size (approximately), the right one is magnified (4 x). The normal colour of the meat is partially masked by red, yellow and green iridescence.
Opal is amorphous, i.e. noncrystalline silicon dioxide containing few percent of water. It is more or less milky and transparent. Precious opal shows variable colours inside. It consists of closely packed silica spheres of uniform diameter, so that the inhomogeneities form regular lattices, as can be seen in the REM pictures supplied by the Mineral Spectroscopy Server of the
California Institute of Technology: opal_gem, opal-beads. The spaces between the beads are filled with water or water-rich silica. Each of them scatters some light.
The scattering centres located in a plane behave very similar to a weakly reflecting plane, and three-dimensional lattice behaves like a stack of such planes. The situation is analogous to that of X-ray diffraction by crystals, only the scale is different. At certain angles, light with a certain wavelength is reflected (according to Bragg's Law of diffraction).
A small piece of raw opal from Ethiopia viewed at different angles. The width of each image is 15 mm.
Structural colours:
Lustre and colours produced by special regular structures can often be found on birds and insects. The colours depend noticeably on the angle of observation, sometimes they can be seen only within a quite narrow region of the viewing angle.
Birds:
Left: A peacock's feather in natural size. Right: A mallard drake (Anas platyrhynchos L.). The colours of the peacock feather and of the mallard's green head and blue speculum are not caused by pigments of the respective colour, but result from the microstructure of the feathers, see below.
Flies and wasps:
Greenbottle Lucilia caesar (L.), a blow-fly of approximately the size of the housefly, and the larger Bluebottle Calliphora vomitoria (L.).
Left: Gadflies like the deer fly Chrysops relictus Meigen have beautiful eyes – right: Gold wasp Chrysis ignita (Linnaeus)
Damselflies:
Banded demoiselle Calopteryx splendens (Harris), left: male
, right: female
Left: The dung beetle Geotrupes vernalis (L.), in particular its under side, shows beautiful lustre. (Length ca. 15 mm.) Right: Leaf beetle Chrysolina fastuosa (Scopoli), length ca. 5 mm.
Butterflies
 |
 |
|
Blue Polyommatus icarus (Rottemburg). Wingspan ca. 32 mm. | Green Hairstreak Callophrys rubi (Linnaeus), Wingspan ca. 30 mm. |
The structures
Butterfly wings are covered with microscopic, partially overlapping scales. The iridescent scales of several species of butterflies and moths have been investigated by electron microscopy; results may be found in the web.
Comparatively simple are the scales of the magnificent, day-active moths of the genus Urania.
 |
Cross section of an iridescent scale of Urania sp. (Source: pages of the School of Physics at The University of Exeter) |
| Urania riphaeus from Madagaskar. (Source: Internet) | |
Four transparent layers of chitin separated by air filled gaps – this corresponds well to the model calculation image shown above.
The iridescent scales of Morpho are more complex. They show thin parallel strips at their upper side. But only the electron microscope reveals the cause of iridescence, namely the particular structure of these strips which are shown in the picture below in cross section.
 |
 |
| Morpho rhetenor from Ecuador. Specimen of the Insects' Museum in Wunstorf/Steinhude, wingspan ca. 15 cm.
| Part of a cross section of an iridescent scale of Morpho aega. Elektron microscope picture: W. Lippert. (From: Karl Gentil, Interferenzfarben ... , Zeiss Informationen 49, p. 80–81, about 1966) |
The image to the right shows an older photograph. Possibly the structure has been slightly deformed in the preparation of the slide.
Nowadays there are other pictures available, e.g. from the School of Physics of the University of Exeter.
Each of the strips consists of a regular arrangement of thin transparent chitinous layers. Altogether these form an array of reflecting surfaces, and the considerations are similar to those sketched in the case of opal. For a more thorough discussion, see this OPOD of the atmospheric optics site.
Iridescent birds' feathers also have submicroscopic regular arrays of fibers or rods which perform like three-dimensional gratings. Peacock's feathers have been investigated by Zi et al. 2003, who found that the cortex of the barbules contain regular arrays of melanin rods embedded in the keratin matrix.
The forewings (elytra) of lustrous beetles have also been studied. For example the leaf-beetle Plateumaris sericea (L.) which occurs in various colours. A simple sequence of layers at the surface: three layers with high refractive index, between them thicker layers with lower index. The layers between 50 and 120 nm thick in one special case (Hariyama et al. 2002).
Tropical jewel beetles of the species Chrysochroa raja have been investigated by Noyes et al. (2007) who found multilayer structures with alternating index of refraction.
Left: Jewel beetle Belionota sp. from Nias/Indonesia, ca. 2.6 cm, Middle: Jewel beetle Megaloxantha nishiyamai from Mindoro/Philippines, ca. 6.5 cm, Right: Plusiotis resplendens from Panama, ca. 2.6 cm. Specimens from the Insects' Museum in Wunstorf/Steinhude.
Some beetles create their iridescent colour in a different way. In layers the chitin molecules are ordered parallel to each other and to the surface. From one layer to the next the orientation angle changes by a small amount (like the steps in a corcscrew staircase), leading to a helical structure (similar to the ordering of molecules in a cholesteric liquid crystal).
As the polarizability of the long molecules is different in longitudinal and transverse directions, such an ordering produces a periodic change in the (anisotropic) index of refraction. Therefore, such a substance behaves similar to a stack of layers with alternating refraction index.
There is, however, a remarkable difference: If the order corresponds, say, to a left-handed screw, the reflected light shows left-handed circular polarization (and vice versa). (This can be shown in a simple model calculation which I will also present here.) By analyzing the polarization of the reflected light, one can thus find out which of the two possibilities discussed is realized, see the work of Jewell et al. (2007).
"The fact that these jewel beetles reflect circular polarization was identified in the early 1900s by a Nobel Prize-winning physicist, A.A. Michelson, who hypothesized that the circular polarization might result from a 'screw structure' within the insect's cuticle, but he did not elaborate on it further."
(from Sharma et al. (2009):
"Structural Origin of Circularly Polarized Iridescence in Jeweled Beetles", see abstract).
Photonic crystals
One-, two- or three-dimensional periodic arrays of refracting or absorbing elements in a matrix are called photonic crystals. (These arrays show band structures for light waves = photons which are similar to the energy bands of electrons = matter waves in crystals, hence the name.) One-dimensional stacks of dielectric layers on glass are used as dielectric mirrors and colour filters. Two- and three-dimensional photonic crystals are fields of research in nanotechnology and could perhaps soon find practical applications.
Back to the index page "the origins of colour"
)
)
